Question: Simplify and expand the following expression: $ \dfrac{n - 3}{3n - 10}+\dfrac{2n}{n + 7} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(3n - 10)(n + 7)$ Multiply the first term by $\dfrac{n + 7}{n + 7}$ $ \begin{align*} \dfrac{n - 3}{3n - 10} \times \dfrac{n + 7}{n + 7} & = \dfrac{(n - 3)(n + 7)}{(3n - 10)(n + 7)} \\ & = \dfrac{n^2 + 4n - 21}{(3n - 10)(n + 7)}\end{align*} $ Multiply the second term by $\dfrac{3n - 10}{3n - 10}$ $ \begin{align*} \dfrac{2n}{n + 7} \times \dfrac{3n - 10}{3n - 10} & = \dfrac{(2n)(3n - 10)}{(n + 7)(3n - 10)} \\ & = \dfrac{6n^2 - 20n}{(n + 7)(3n - 10)}\end{align*} $ Now we have: $ = \dfrac{n^2 + 4n - 21}{(3n - 10)(n + 7)} + \dfrac{6n^2 - 20n}{(n + 7)(3n - 10)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{n^2 + 4n - 21 + 6n^2 - 20n}{(3n - 10)(n + 7)} $ $ = \dfrac{7n^2 - 16n - 21}{(3n - 10)(n + 7)}$ Expand the denominator: $ = \dfrac{7n^2 - 16n - 21}{3n^2 + 11n - 70}$